What is the distance formula between two points?

The distance between two points (x₁, y₁) and (x₂, y₂) is: d = √((x₂-x₁)² + (y₂-y₁)²). This comes from the Pythagorean theorem: the distance is the hypotenuse of a right triangle with legs of length |x₂-x₁| and |y₂-y₁|.

What is the difference between Euclidean and Manhattan distance?

Euclidean distance is the straight-line ('as the crow flies') distance: √((x₂-x₁)² + (y₂-y₁)²). Manhattan distance only allows movement in horizontal and vertical directions (like city blocks): |x₂-x₁| + |y₂-y₁|. Euclidean is always ≤ Manhattan distance.

How do I calculate distance from latitude and longitude?

Use the Haversine formula: a = sin²(Δlat/2) + cos(lat1)×cos(lat2)×sin²(Δlong/2); d = 2×R×arcsin(√a), where R = 6,371 km. This gives the straight-line great-circle distance. For example, New York (40.71°N, 74.01°W) to London (51.51°N, 0.13°W) ≈ 5,570 km.

How do I find the midpoint between two coordinates?

The midpoint between (x₁,y₁) and (x₂,y₂) is ((x₁+x₂)/2, (y₁+y₂)/2). For 3D points (x₁,y₁,z₁) and (x₂,y₂,z₂): midpoint = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).

Distance Calculator

The Distance Calculator handles three types of distance problems: 2D Coordinates for geometry and graph problems using the Pythagorean theorem (also shows Manhattan distance, midpoint, and slope), 3D Coordinates for three-dimensional space problems (Euclidean distance + midpoint), and Lat/Long (Geographic) for real-world map distances between any two locations using the Haversine formula — the same method GPS devices use to calculate straight-line distance. Results are shown in both kilometers and miles for geographic mode, and the unit you enter for coordinate modes.

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calendar_today Updated: April 2026

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Point A

x₁

y₁

Point B

x₂

y₂

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Enter two points to calculate distance

functions Distance Formulas

2D Euclidean

d = √((x₂−x₁)² + (y₂−y₁)²)

3D Euclidean

d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)

Manhattan (L1)

d = |x₂−x₁| + |y₂−y₁|

Midpoint

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Earth Radius (R)

6,371 km · 3,959 mi

public Lat/Long Quick Reference

1° latitude ≈ 111 km / 69 mi

1° longitude @ equator ≈ 111 km

1° longitude @ 45° ≈ 79 km

NY → London ≈ 5,570 km

Earth circumference 40,075 km

lightbulb Quick Tips

•Euclidean ≤ Manhattan — straight line is always shortest
•Lat/Long gives great-circle (as-the-crow-flies), not driving distance
•Negative latitude = South; negative longitude = West
•Use decimal degrees: 40°42′46″N → 40.7128

How to Use This Calculator

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Choose Distance Type

Select 2D Coordinates for flat-plane geometry, 3D Coordinates for space problems, or Lat/Long for real-world geographic distances.

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Enter Point Coordinates

Input the x and y values (and z for 3D) for both points. For geographic mode, enter latitude and longitude in decimal degrees.

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Get Your Distance

See the straight-line distance instantly. 2D mode also shows Manhattan distance and the midpoint coordinates.

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Check the Formula

The step-by-step calculation is shown below the result so you can verify the math or learn how it was computed.

The Formula

The 2D distance formula is a direct application of the Pythagorean theorem in a coordinate plane. For 3D, an extra squared difference term is added under the radical. The Haversine formula handles the curvature of the Earth, giving the shortest great-circle distance (as-the-crow-flies) between two geographic points.

2D: d = √((x₂-x₁)² + (y₂-y₁)²) | 3D: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²) | Haversine: d = 2R·arcsin(√(sin²(Δφ/2) + cos φ₁·cos φ₂·sin²(Δλ/2)))

lightbulb Variables Explained

(x₁,y₁), (x₂,y₂) Coordinates of the two points in 2D space
(x₁,y₁,z₁), (x₂,y₂,z₂) Coordinates of the two points in 3D space
φ (phi) Latitude in radians
λ (lambda) Longitude in radians
R Earth's mean radius ≈ 6,371 km
Δφ, Δλ Difference in latitude and longitude between the two points

tips_and_updates Pro Tips

The Euclidean distance is the 'straight-line' distance — the shortest possible path between two points.

Manhattan distance (also called taxicab or L1 distance) counts only horizontal + vertical movement — useful for grid-based problems.

For latitude/longitude: 1 degree of latitude ≈ 111 km (69 miles). Longitude degrees vary with latitude.

The Haversine formula gives the great-circle distance — the shortest path over the Earth's surface, not driving distance.

For 3D distance: think of it as applying the Pythagorean theorem twice — once in the XY plane, then including the Z dimension.

Frequently Asked Questions

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